Circular Lending: Ouroboros Schemes

In a financial system without supervision leverage can grow dramatically.  Let's explore the impact of rehypothecation and lending haircuts on real balance sheets.  In this example we refer to USD as the currency.  That can be any stablecoin (or actual USD) it doesn't matter here.

Start with a BTC spot of 60k and an overcollateralization haircut of 25%.

Our "lender" begins with the following balance sheet:

The lender borrows 1M USD, in cash, and starts there.  After making a single collateralized loan for 1 BTC they have:

The borrower has 48k in their bank account.  As long as BTC stays above 48k -- where the liquidation occurs -- this transaction is on snooze mode.  Let's say the borrower goes and buys 48k worth of BTC.  That is buying pressure of 0.8 coins.  Now pledge those coins to another lender.  The two lender balance sheets are:

Of course the next step is to buy 0.64 coins and pledge them to someone else:

The aggregate system now has:

That is not crazy.  If we allow the buy-pledge loop to run forever through these 3 banks we can generate more loans.  And that generates an aggregate balance sheet of:

The total buying here is 4 coins.  And we have 240k of loans outstanding against an initial collateral value of 60k.  This is unsurprising as max leverage is just 1/haircut.

Over time, until there is a crisis, we know lending standards get looser.  Let's say that is reflected in lower haircuts.  At 10% it is useful to compare the "3 loans" and "infinite loop" aggregate balance sheets.  Here we are still running the loans through a total of 3 banks -- total funding is still 3M:

The situation with 1 loan for each lender is pretty mild.  But the infinite loop aggregate system is far more levered!  This activity generated buying of 10 coins and leaves behind a liquidation order for 11.

With a haircut of 2%:

Now we are able to exhaust a system-wide total of 3M lending against a single coin.  This process generates buy orders for 50 coins.  And the aggregate liquidation order is for 51x the initial collateral.

Safety of the loans isn't even the primary concern here.  This activity facilitated buying 50 more coins.  This isn't someone trading 50x on an exchange -- each individual transaction here is just an at-market loan and a cash purchase of coin.

We don't even require the same person to keep taking out these loans.  The simplest sequence of events is likely:

1. Borrow against BTC
2. Buy BTC with loan proceeds
3. Deposit BTC into a yield generating protocol
4. Protocol lends the BTC

That is a circle of collateralized.  This is not a ponzi scheme and it's not a pyramid scheme.  It's a long loop of borrow-buy-borrow-buy until the marginal loan is too small to bother.  And it generates an awful lot of leveraged buying.

This does not happen in the regulated financial system because aggregate system balance sheet leverage is constrained.  Obviously problems can, and have, occurred with excessive leverage.  But each balance sheet is supervised by internal agents (i.e. the risk department), external agents (i.e. the bank's investors) and the regulators (i.e. banking, markets, securities, whatever).

It's a big ugly complex system.  It's not perfect.  But it is fundamentally different from a collection of  smart contracts unconsciously running through that loop.

At a high level it is an empirical question which is safer.  These systems have very different properties.  It is, however, quite clear that loose lending in DeFi can directly lead to large coin purchases and leave behind large liquidation orders.

Each player has their own defensive arrangements in place.  Without the possibility of extraordinary liquidity assistance it feels as though everyone will be forced to turn their cards over at once.

We propose the name "ouroboros scheme" for this configuration.  As long as the underlying asset retains adequate value this can continue forever.

Scheme is meant in the British sense that does not connote something negative -- not the American sense which does carry a suggestion of something sinister.  In the UK it is totally normal to have a "pension scheme."  This term would scare Americans and many other English-speaking groups.  Divided by a common language indeed.

How Big Can Collateralized Crypto Lending Get?

BTC is in fixed supply.  This is one of, if not the, top reasons people who love it love it.  However this fixed supply has interesting implications for the collateralized crypto lending markets.

As discussed previously on this blog if we start with 1 unit of stablecoin to lend out against BTC collateral and require an overcollateralization haircut of X% we can end up with 1/X in gross loans (longer discussion here).

We know stablecoins are printed and lent to various platforms from the public press.  Here we will discuss how the fixed supply of BTC constrains the growth of this lending.

The key observation here is: the aggregate liquidation order has to be for fewer coins than the total outstanding.  Simply put the price must be high enough that: \begin{equation} \frac{Stablecoins}{Haircut} = Supply \times Price \end{equation}

Assuming 18M BTC outstanding we get the following minimum prices to support lending at the given haircut:

At an average haircut of 10% with 50B raw stablecoin outstanding to lend the market requires a BTC price at least 27.8k.  At 4% and 75B the required floor is now over 100k.  With higher haircuts the prices are obviously lower.  This is not a difficult spreadsheet to build.

Take a look at the bottom few rows.  This is a real limit on non-fiat-backed stablecoin issuance!  In some sense a lower haircut is like a looser monetary policy -- lending is easier.  But with a fixed money supply the price must rise, possibly dramatically, to provide adequate collateral for non-fiat-backed loans.

A stablecoin issuer that wants to issue more coins against collateral must make a choice: how do they balance the haircut against the number of coins they want (need) to issue?  Note that when the haircut is below 50% most of the lending is not from the original issuer -- it is instead follow-on rehypothecation loans out there in defi land.

A lender that wants to ramp up business has to monitor what fraction of the total coins they might already need to sell.  The supply of coins is currently growing slowly.  But in the short term -- in the sort of timeframe one might need to execute a liquidation order -- the supply is effectively fixed.

There Is No Free Lunch

Printing coins to push up prices, or to grease the wheels of lending platforms, or for any other purpose may at first feel like an unconstrained activity, something that can go on forever.  But this is ultimately a markets activity.  We are talking about borrowing and lending, buying and selling.

Lending against collateral is a transformation of some sort.  It is an event creating a flavor of credit.  Otherwise what purpose does it serve?  Creating credit in a system with a fixed-in-stone money supply feels like an activity that cannot continue without limit.

This is that limit: there is a price at which more than the entire monetary base needs to be liquidated to satisfy someone's liabilities.  And these liquidations may occur on platforms nobody can easily shut down.

Crypto Lending, Market Structure and Stop-Losses

Crypto lending is now a "big thing."  DeFi protocols, centralized services, stablecoin loans -- these are all very much in the news.  All of this activity is unregulated in the sense that leverage is only limited by the risks involved parties are willing to take.  Given the size and evident high risk tolerances of many participants in the space it's worth asking just how much leverage is floating around.

Market Setup

Let's start with two assumptions to simplify things:

1. Loans are of stablecoin (STA) with crypto collateral (COL).
2. At inception loans are X% overcollateralized with a single margin call when the value of the collateral drops to the balance of the loan.

These are broadly in line with the mid-2021 crypto markets.  The most common case looks to be USD-something lent against BTC with an X of, say, 30%.

How Much Gross Lending Is There?

Within this structure it is perfectly plausible for a borrower to use their STA loan proceeds to buy more COL and then re-pledge.  Given the lack of market-wide supervision and high risk tolerances it is reasonable to believe this kind of leveraged buying is normal.  But how much can there be?

The math is straightforward.  The first loan is for 1.  Let's assume that gets converted into COL without any fees.  The second loan is then for \begin{equation}\frac{1}{1+X}\end{equation}  And the third is for \begin{equation}\frac{1}{(1+X)^2}\end{equation}  If this process loops forever the total is \begin{equation}\sum_{i}{\frac{1}{(1+X)^i}} = \frac{1}{X}\end{equation} 

Note this takes the initial loan's proceeds as 1 rather than the initial value of the collateral. It's cleaner that way and doesn't impact the high level conclusions.  In fact it may more closely approximate the real world as discussed below.

Margin Calls and Liquidations

If all of this borrowing and lending happens at some price P (STA per COL) all the margin calls occur if P falls X%.  This is obviously simplistic but it can still be instructive.

There is an obvious relationship among the degree of overcollateralization, the liquidation level and the size of the liquidation.  Again assuming everything occurs at the same initial levels we have a liquidation at P/(1+X) for 1/X units.

Intuitively this makes sense: a lower X gives us a larger and closer stop-loss order.  Lower X = higher leverage.  This is a boring result.

Leveraged Buying

The giant stop-loss order is a sign a bunch of leveraged buying must have occured!  This is also wholly unsurprising.  Each incremental conversion of STA loan proceeds to COL is a crypto purchase.  In total we also have leveraged buying of 1/X units of COL!

Stablecoin lending of this type, even with 100% backed stablecoins, can still generate a lot of upward pressure on prices given a sufficiently-well-functioning lending ecosystem.

This is a classic feature of traditional financial markets -- and a key reason for leverage supervision.  Note that absolutely nothing dodgy is happening here.  This may or may not represent more risk than the system can bear.  But no individual actor is doing anything dishonest.

Also note there is no reason to believe sufficient backing exists for STA to liquidate COL into USD at it's price after the leveraged buying!  The quantum of leveraged buying is entirely dependent on X.  Presumably the price post-buying is therefore at least weakly dependent on X.  But this has nothing to do with the quantity of STA outstanding.  STA can be completely covered by USD reserves, and STA redemption itself be free of risk, but COL still cannot be liquidated into USD at the higher price given only STA's reserves.

Stablecoin Issuers

What if STA is also issued in this fashion, with overcollateralized COL as backing?  Several major stablecoin issuers look to provide such loans.  This depends a bit on what you think "approved investments" and similar phrases mean precisely but we are simplifying on purpose here.

If Y% of STA that's been issued is backed by COL then we only have USD backing to liquidate (1-Y)% of STA prior to any collateral sales.  But selling COL for STA does not increase the balance of reserves!  Raising reserves would require liquidating collateral in the COL-USD market.

Plugging In Numbers

Let's continue to use 30% for X.  How much lending is there?  Well one good place to start is the balance of stablecoin issuer non-USD-equivalent holdings.  Let's take total stablecoin issuance at USD 100B and say a third is this type of lending.  And all against BTC.  This is a simplification.  It's a basic model for stylized answers.  Whatever.

This gives us total outstanding lending of USD 110B.  At a price of 65k that is a stop-loss for 110B at 50k.

Also note that this assumes cash backing of 50B.  There are not enough in the cash reserves to support this stop-loss order against USD even ignoring all other backed tokens.

The fragility of this system depends on X and Y.  A lower X raises the risk.  And a higher Y raises the risk.  If Y is 0 the system still may not be able to convert all the tokens into hard currency at the prevailing price.

Extreme Scenarios

To illustrate plausible states of the world let's look some extremes, again starting from a 65k price.

First take X as 66% and Y as 10%.  As a market-wide average this feels pretty conservative.  In this case we have:

• 90B in USD reserves
• 10B in initial loans, 15B in total loans
• A stop-loss for 15B at 39k

Note that reserves are insufficient but barely.  This likely doesn't scare anyone.  Almost certainly it doesn't scare the median market participant.

At the other extreme take X as 10%, Y as 80%.  Surely some lending is occuring at these kinds of levels but this is quite aggressive for the average.  Anyway:

• 20B in USD reserves
• 80B in initial loans, 800B in total loans
• A stop-loss for 800B at 59k

That stop-loss requires liquidating 13.5 million coins -- this is obviously a total train wreck.  You can find plasuible-but-too-aggressive sets of parameters where the total ecosystem market cap is insufficient to cover the stop-loss.

If X is 5% and Y is 90% then the balance of outstanding loans would be 1.8T.  A stop-loss to raise 1.8T at 61.9k is impossible -- that would require selling 29 million coins.

Perhaps the real leverage limit in this market will prove to be the capped issuance.  Any lender that sees stop-losses for 10%, or 20% or 50% of total coins outstanding knows they have a problem!

Spoiler alert: the inability to inject temporary liquidity is a known problem with fixed exchange rate regimes.